On the eventual absorption of a quantum particle

A NNALES DE L ’I. H. P., SECTION A

L. C. T ^HOMAS

On the eventual absorption of a quantum particle

Annales de l’I. H. P., section A, tome 21, n

^o

1 (1974), p. 81-87

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81

On the eventual absorption

of

a

quantum particle

L. C. THOMAS

Department ^{of Pure} Mathematics, University College ^of Swansea, Singleton Park, Swansea

Ann. Inst. Henri Poincare, Vol. XXI, ^n° 1, 1974,

Section A :

Physique thearique.

ABSTRACT. - For ^a

general system consisting

^{of a}

quantum particle

and an

absorbing detector,

^the

probability

that the detector will

eventually

absorb the

particle

^is calculated.

RESUME. 2014 Pour un

systeme, qui

^{est une}

particle quantique

^{et un instru-}

ment

absorbant,

^on obtient la

probability

que I’instrument absorbera la

particle.

1. INTRODUCTION

Quantum

stochastic _processes

[1]

^{enable one to} deal with

quantum systems

^{in which the time evolution is not}

unitary.

^In

particular,

^{one can}

consider a

system consisting

^{of a}

quantum particle

^{with an}

absorbing detector,

^which will absorb the

particle

^{when it comes in contact with}

it,

as for

example

ⁱⁿ

[2] [3].

^The

particle’s

space of ^states is a

separable

Hilbert _space, with a

densely

defined Hamiltonian

H, describing

^{its time}

evolution. We assume that the

spectrum

^{of H is}

only

pure

point,

^{and its}

eigenvectors

e~, of

eigenvalue

,u~, form an orthonormal basis. For the combi- ned

system,

the space of

possible

^{states is}

~(~f),

^the

symmetric

Fock space built on and let

R

be the extension of H from Jl’ In the termi-

nology

^of quantum stochastic processes the ^state space

(V, !)

^{is taken}

as V ⁼ the Banach space of self

adjoint

^{trace class}

operators

on with! the trace. The detector is

represented by

^{the canonical}

Annales de l’Institut Henri Poincaré - Section A - Vol. XXI, ^n° 1 - 1974 ^.

82 ^{L. C. THOMAS}

annihilation operator

A(d), d being

^an element of

~f,

^{which is}

supposed

to describe the characteristics of the detector.

The

quantum

stochastic process for this

system

^{is a set} of instrument

~t,

indexed

by

^{a time}

parameter t

^E

[0, oo),

^{each of which} maps the initial

state of the

system

^{onto the state at} time t,

dependent

upon what has

happened

^{up to time t.}

Formally

^{~’t : V x}

V,

^where

Xt

^{is the}

sample

space,

consisting

^{of all}

possible

^events that could have

happened

^{to the}

detector _up to time t. Thus

X, = {

^{s : S E}

^[0, t] ~

^{u z,} since either the

particle

has been absorbed at a time s less than t, ^{or no}

absorbing

^has

yet

^occurred the

point

^z. The definition of such instruments was

given

ⁱⁿ

[1],

^{where it}

was shown that

Stv

^{= v E}

V,

^{the states} conditional on no

hitting

form a

semigroup

^{and if we}

writer

^{= then the no}

hitting

^{time deve-}

lopment

^{is a}

perturbation

of the usual time

development, given by

Let the initial state of the

particle

^be

P Ð,

^the

projection operator

^{onto the}

subspace

^of

spanned by

cI&#x3E; ⁼

(0, ~ 0,

⁰

...).

^The

system

^{is then} described

by H, ~ and ~

^{and the}

probability

^{that no}

absorption

^{has occurred}

b

^{time r is}

while in

[1],

^it is shown that the

probability

^{that the} detector absorbs the

particle

ⁱⁿ the interval

(r,

^{t +}

bt)

^is

We are interested in what is the

relationship

^between

H, 1~ and d,

^{in order} that with

probability

one, the detector will

eventually

^{absorb the}

particle.

We will show that the

particle

^will

eventually

be absorbed if and

only if,

when d and h are

expanded

^{in terms of} the orthonormal basis

consisting

of

eigenvectors

^of

^{H, d}

^has a non-zero

component

^{of all the}

eigenvectors

contained in the

expansion

of h. More

formally

There ^" is a

slight ambiguity

^{in the statement of the}

proposition

^{since the}

eigenvectors

^for

degenerate eigenvalues

^{are ~ not defined}

uniquely.

^However

Annales de 1’Institut Henri Poincaré - Section A

83

ON THE EVENTUAL ABSORPTION OF A QUANTUM PARTICLE

to avoid this

ambiguity,

^if the initial

one-particle

^{state is}

h,

^{we shall}

always

assume that one of the

eigenvectors

^{in the}

eigenspace

^is the normalised

projection

^{of h onto that}

eigenspace,

and the others are

orthogonal

^{to this}

vector.

2. PROOF OF PROPOSITION

Since ~ has

only

^a

1-particle component

^and

B,

leaves this

space invariant,

it is sufficient to restrict attention to the

1-particle

^{space, ~f.}

where

Pd

^{is the}

projection

^{onto the}

subspace spanned by

^{d. Let}

y =F O}

^and

~2 = O}

^where

V { }

implies

^the closed linear

subspace spanned by

the elements

within { }.

The

proposition

^is

proved

^{if we can show that}

B,

^{leaves the}

~t invariant,

is

strongly

contractive on i. e. _{t -~} lim

Bth

^{= 0 b’h E and is}

unitary

00

o n

~2 .

where

Since

Pd 1.Jt’2

^{= 0 and}

since e-tHt

leaves

H2

^invariant

Cm1 1£2

^{= 0. Thus}

and so is

unitary.

^{It is now} obvious that

Bt

^leaves

~2

^invariant.

Let

hl

^E

~1

^{and let} e~2 be ^an

eigenvector

^{of H in then}

This is true for all et2 ⁱⁿ

~2

^{and so}

Bt

^leaves

~1

^invariant.

Finally

it is necessary ^to show that

Br 1Jf’1 is strongly contracting.

^Davies

[1]

has shown that such a

semigroup

^{is a} contraction

semigroup,

^{so we need}

1

only

^{show that -}

iH - 2 ^Pd

^{has no pure}

imaginary eigenvalues.

^Consider

the action of this operator ^on the

eigenvectors

e~, which _span

:Ýf1

Choose the

phase

factors of

the ej,

^so

^{that (}

is real and

the dj

^{’s are}

Vol. XXI, ^n° 1 - 1974.

84 L. C. THOMAS

real

with ) ~

⁼ l. The matrix for

( - ^{iH - ~} with

this basis is

}

^{B 2}

^/~~

of the form

and the

eigenvalues

^of the operator ^are

given by

Put ~, ⁼ then

If the operator has ^a pure

imaginary eigenvalue,

^then there exists a real value

of

^{for which and}

therefore |f( )|2

^{is zero.} However 2 . 7 is the sum of two squares which have non-coincident roots

(since

^the

~

must be

non-zero).

The result is thus _proven.

1’Institut Poincaré - Section A

85 ON THE EVENTUAL ABSORPTION OF A QUANTUM PARTICLE

3. ALTERNATIVE APPROACH

One is able to

get

^{more detailed} information about the

probability

^of

absorption

^{of the}

particle

^{at any}

^time, by using

^an

expansion

^{formula of}

the form 2.2 for

B,

^on

^~~_~),

where

Since 03A6 has

only

^a

one-particle component

^is

only

^non-zero

in the

O-particle

state, ^so

where is ^a function.

We assume for

simplicity

that d is such that

only finitely many dk

^{’s are}

non-zero. Since 03A6 =

(0, h, 0, ... )

^and

using

^the

expansions

^{of d and h}

of

Proposition 1,

Taking Laplace transforms,

^we

get

In

[1],

^{it was shown that the}

probability

that the detector will absorb ^o the

particle

^~ in the interval

(t,

^{t + is}

Vol. XXI, n° 1 - 1974.

86 L. C. THOMAS

Thus to calculate the

density

function of the

absorbing time,

^{one takes the}

inverse

Laplace

^{transform of}

~(p),

^to

obtain a(.s) 12.

^The

probability

^that

the detector will

eventually

absorb the

particle

^is

roo ^{I 2d.s.}

^Plan-

cherel’s formula for

Laplace

transforms can be

applied

^{in this case, since}

is

obviously

^a

^L2

functions. So

where x is real. In section

2,

we have shown that the

probability particle

is

eventually

^{absorbed is}

Rewriting

this result in terms of

the p

^and

using 3. 7,

it follows that the functions of 3 . 5 must form an ortho-normal

system,

^{i. e.}

We

again

^choose

the ei

^so

that dk

^is real for all

/,

^{and let}

03BBk

⁼⁼

d2k

^{so = 1.}

Proposition

^{1 thus}

implies

^{that if we have a finite set of}

positive

^numbers

/, ⁼⁼ 1 ⁼ N

with 03A303BBk

⁼⁼

^1.

^{then the functions}

" ,

^{, with}

weight

z-j

~+~

~

function

form an orthonormal set, i. ^e.

ACKNOWLEDGMENTS

My

^{thanks are due to Dr E.} B. Davies both for

suggesting

^this

problem

and for several valuable conversations

concerning

^it.

Annales de l’Institut Henri Poincaré - Section A

ON THE EVENTUAL ABSORPTION OF A QUANTUM PARTICLE 87

[1] ^{E. B.} DAVIES, Quantum stochastic processes I. Comm. Math. Phys., ^t. 15, 1969,

p. 277-304.

[2] ^{E. B.} DAVIES, Quantum stochastic processes III. Comm. Math. Phys., ^t. 22, 1970,

p. 51-70.

[3] ^{L. C.} THOMAS, Probabilistic calculations on the ground ^state of the harmonic oscillator.

J. Statist. Phys., ^t. 7, 1973, p. 287-294.

(Manuscrit reçu Ie 4 ^’ octobre 1973)

Vol. XXI, ^{n° 1 - 1974.}